A game of digits and seventh powers
Source: Taiwan 2014 TST3 Quiz 1, P2
July 18, 2014
modular arithmeticnumber theoryrelatively primenumber theory proposedprimitive root
Problem Statement
Alice and Bob play the following game. They alternate selecting distinct nonzero digits (from to ) until they have chosen seven such digits, and then consider the resulting seven-digit number by concatenating the digits in the order selected, with the seventh digit appearing last (i.e. ). Alice wins if and only if the resulting number is the last seven decimal digits of some perfect seventh power. Please determine which player has the winning strategy.